- Damien Roy: droy@uottawa.ca
- Daniel Fiorilli: Daniel.Fiorilli@uottawa.ca
- Saban Alaca: SabanAlaca@cunet.carleton.ca

**Speaker:**Lucile Devin (U. d'Ottawa)**Title:**Generalizations of Chebyshev's bias.**Abstract:**Following ideas of Rubinstein, Sarnak and Fiorilli, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions L(s) = ∑_{n≥1 }λ_{f }(n) n^{-s}satisfying natural analytic hypotheses. We put the emphasis on weakening the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular we establish the existence of the logarithmic density of the set {x ≥ 2 : ∑_{p≤x }λ_{f }(p) ≥ 0} conditionally on a much weaker hypothesis than was previously known. We include applications to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases.

**Speaker:**Luca Ghidelli (U. of Ottawa)**Title:**Large gaps in the values of positive-definite cubic and biquadratic diagonal forms**Abstract:**Let S be the set of the natural numbers that can be written as a sum of 3 nonnegative cubes, or more generally are the values of a given (homogeneous, positive-definite, diagonal) polynomial F of degree d in d variables. The distribution of such a set S in N is largely conjectural, but in some cases we can prove that there are arbitrarily large intervals contained in the complement, thanks to a special phenomenon in low degree. The proof involves some beautiful mathematics, from Hecke L-functions to the theory of Kummer extensions.

**Speaker:**Daniel Fiorilli (U. d'Ottawa)**Title:**Low-lying zeros of quadratic Dirichlet L-functions: the transition**Abstract:**I will discuss recent joint work with James Parks and Anders Södergren. Looking at the one-level density of low-lying zeros of quadratic Dirichlet L-functions, Katz and Sarnak predicted a sharp transition in the main terms when the support of the Fourier transform of the implied test functions reaches the point 1. By estimating this quantity up to a power-saving error term, we show that such a transition is also present in lower-order terms. In particular this answers a question of Rudnick coming from the function field analogue. We also show that this transition is also present in the Ratios Conjecture's prediction.

**Speaker:**Nikolay Moshchevitin (Lomonosov Moscow State University)**Title:**Inequalities between different Diophantine exponents**Abstract:**We will consider different Diophantine exponents for simultaneous Diophantine approximation and for one linear form and discuss some relations between them. Some of the inequalities are optimal. This follows from D. Roy's theorem about Schmidt-Summerer's graphs. Some of the results are obtained jointly with A. Marnat.

**Speaker:**Nathan Ng (Lethbridge U.)**Title:**The sixth moment of the Riemann zeta function and ternary additive divisor sums**Abstract:**Hardy and Littlewood initiated the study of the 2k-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an asymptotic formula for the fourth moment. In this talk we consider the sixth moment of the zeta function on the critical line. We show that a conjectural formula for a certain family of ternary additive divisor sums implies an asymptotic formula for the sixth moment. This builds on earlier work of Ivic and of Conrey-Gonek.

**Speaker:**Youness Lamzouri (York U.)**Title:**On the distribution of the maximum of cubic exponential sums**Abstract:**In this talk, we shall present recent results concerning the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry, and can also be generalized to other types of $\ell$-adic trace functions. In particular, some of our results also hold for partial sums of Kloosterman sums. As an application, we exhibit large values of partial sums of Birch sums, which we believe are best possible.

**9:45-10:45: Fabien Pazuki (U. of Copenhagen and U. of Bordeaux),***Regulators of elliptic curves.***Abstract:**In a recent collaboration with Pascal Autissier and Marc Hindry, we prove that up to isomorphisms, there are at most finitely many elliptic curves defined over a fixed number field, with bounded rank at least equal to 4 and bounded Mordell-Weil regulator.**11:00-12:00: Luca Ghidelli (U. of Ottawa),***Gaps in values of diagonal forms and applications to Diophantine approximation.***Abstract:**Consider the set S of natural numbers that can be written as a sum of four 4th powers. We prove that between the values of S there are arbitrarily large gaps, via an argument that involves the primes congruent to 5 modulo 8. Meanwhile we prove, using the Hardy-Littlewood circle method, that "sufficiently often" the gaps between these values are "not too wide". Together, these two facts have arithmetic consequences on the values of a certain "biquadratic theta function".
Abstract:
**12:00-14:15: Lunch****14:15-15:15: Damien Roy (U. of Ottawa),***Simultaneous approximation to exponential of algebraic numbers.***Abstract:**The theorem of Lindemann-Weierstrass asserts that the exponentials of distinct algebraic numbers are linearly independent over the field of rational numbers. The proof uses a construction of simultaneous rational approximations to such exponentials values, which goes back to Hermite. In this talk, we show that, from an adelic perspective, these approximations are essentially best possible. The point of view partly explains the nature of the algebraic numbers whose exponentials have a structured continuous fraction expansion. We conclude with a conjecture regarding simultaneous approximations to such values in adèle rings.**15:30-16:30: Daniel Fiorilli (U. of Ottawa),***Chebyshev's bias in Galois groups.***Abstract:**This work is joint with Florent Jouve. In this talk we will discuss Chebyshev's bias in the distribution of primes according to Chebotarev conditions. For example we will compare the number of primes p congruent to 1 modulo 3 for which 2 is a cube modulo p with those for which this condition does not hold. One of our goals will be to study extreme biases, that is we will give conditions on the implied Galois groups which guarantee significant asymmetries. We will see that those questions are strongly linked with the representation theory of this group. For example, in the S_{n}case we will take advantage of the rich and combinatorial representation theory of the symmetric group - in particular bounds of Roichman, Féray, Sniady, Larsen et Shalev. We will also apply inverse Galois type results.

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